Probabilistic estimation of the Dynamic Wake Meandering

Wind Energ. Sci., 6, 1117–1142, 2021https://doi.org/10.5194/wes-6-1117-2021© Author(s) 2021. This work is distributed underthe Creative Commons Attribution 4.0 License.

Probabilistic estimation of the Dynamic WakeMeandering model parameters using

SpinnerLidar-derived wake characteristics

Davide Conti1, Nikolay Dimitrov1, Alfredo Peña1, and Thomas Herges2

1Department of Wind Energy, Technical University of Denmark,Frederiksborgvej 399, 4000 Roskilde, Denmark

2Wind Energy Technologies, Sandia National Laboratories, Albuquerque, New Mexico 87123, USA

Correspondence: Davide Conti ([email protected])

Received: 31 December 2020 – Discussion started: 4 February 2021Revised: 30 June 2021 – Accepted: 19 July 2021 – Published: 9 September 2021

Abstract. We study the calibration of the Dynamic Wake Meandering (DWM) model using high-spatial- andhigh-temporal-resolution SpinnerLidar measurements of the wake field collected at the Scaled Wind Farm Tech-nology (SWiFT) facility located in Lubbock, Texas, USA. We derive two-dimensional wake flow characteristicsincluding wake deficit, wake turbulence, and wake meandering from the lidar observations under different at-mospheric stability conditions, inflow wind speeds, and downstream distances up to five rotor diameters. Wethen apply Bayesian inference to obtain a probabilistic calibration of the DWM model, where the resulting jointdistribution of parameters allows for both model implementation and uncertainty assessment. We validate the re-sulting fully resolved wake field predictions against the lidar measurements and discuss the most critical sourcesof uncertainty. The results indicate that the DWM model can accurately predict the mean wind velocity andturbulence fields in the far-wake region beyond four rotor diameters as long as properly calibrated parametersare used, and wake meandering time series are accurately replicated. We show that the current DWM modelparameters in the IEC standard lead to conservative wake deficit predictions for ambient turbulence intensitiesabove 12 % at the SWiFT site. Finally, we provide practical recommendations for reliable calibration procedures.

1 Introduction

Wake effects are perceived as one of the largest sources ofuncertainty in energy production and load estimates of on-shore and offshore wind farms (Walker et al., 2016). Withinan iterative design process and/or optimization study, wakeeffects on aeroelastic turbine responses are predicted usingengineering wake models, e.g. the Dynamic Wake Meander-ing (DWM) (Madsen et al., 2010) and Frandsen (Frandsen,2007) models, which can be used within simple and fast de-sign tools (Braunbehrens and Segalini, 2019). Their mainlimitation is their reduced ability to fully resolve the turbu-lence structures of the wake field, which often leads to aninaccurate representation of the flow field and biased powerand load predictions (Reinwardt et al., 2018). To minimizethe modelling uncertainty, it is a common practice to cal-

ibrate engineering wake models using field measurementswhen available or using higher-fidelity simulations like com-putational fluid dynamics (CFD).

Wind lidars have become popular for studying wind tur-bine wakes due to their higher spatial resolution and easeof installation compared to traditional anemometers mountedon meteorological masts (Machefaux et al., 2016). The use oflidar measurements to calibrate low-order wake models hasalready been successfully adopted (Trabucchi et al., 2017;Reinwardt et al., 2020; Zhan et al., 2020a). Although high-quality lidar observations of the wake field are available(Käsler et al., 2010; Iungo et al., 2013; Aitken et al., 2014),the spatial and temporal resolution required to characterizewake deficit, wake turbulence, and meandering characteris-tics is rarely achieved. Such resolution is a key characteristicfor the development and evaluation of dynamic wake models.

Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.

1118 D. Conti et al.: Probabilistic estimation of the Dynamic Wake Meandering model parameters

The Scaled Wind Farm Technology (SWiFT) experiment,conducted at Sandia National Laboratories between 2016and 2017 (Herges et al., 2017, 2018; Herges and Keyantuo,2019), provides a fairly complete and suitable dataset for thecalibration and evaluation of wake models (Doubrawa et al.,2019, 2020; Conti et al., 2020a). The SWiFT dataset con-sists of concurrent measurements of inflow conditions froma heavily instrumented meteorological mast, high-spatial-and high-temporal-resolution measurements of a single wakeflow field behind a turbine from a nacelle-mounted Spinner-Lidar, and power and load measurements from a second tur-bine operating in the waked field. The detailed instrumen-tation of the site allows the investigation of the wake fieldvariability under different atmospheric-stability conditionsas well as the analysis of the wake-induced effects on thewaked-turbine operation (i.e. power and load predictions).

Here, we analyse the SWiFT dataset aiming at calibrat-ing and evaluating the DWM model. This model is recom-mended in the IEC 61400-1 standard (IEC, 2019) for thepurpose of wind turbine and wind farm design certification,and it is widely used in load assessments under wake con-ditions (Larsen et al., 2013; Galinos et al., 2016; Reinwardtet al., 2018, 2020; Dimitrov, 2019). The DWM model simu-lates wind field time series and is divided into three parts: awake deficit component, which simulates the velocity deficit;a wake-added turbulence component; and a wake meander-ing component, which is a stochastic meandering process.These three components are presumed to affect wind turbineloading conditions (Keck et al., 2012; Galinos et al., 2016;Larsen et al., 2013; Dimitrov, 2019). Although several stud-ies have demonstrated the superior performance of the DWMmodel compared to other engineering wake models that onlypredict steady wake features (Thomsen et al., 2007; Larsenet al., 2013; Reinwardt et al., 2018), the accuracy of both theDWM-simulated wake flow fields and the resultant turbinepower and load predictions is still to be assessed.

1.1 A review of the DWM model

The underlying hypothesis of the DWM model is to considerthe wake as a passive tracer of the large incoming turbulencestructures. The so-called split-in-scales assumption (Larsenet al., 2008) states that the large-scale turbulent eddies con-tained in the atmospheric boundary layer are the main driversof the wake meandering, whereas the smaller turbulent ed-dies govern the wake deficit evolution downstream of the ro-tor. Further, wake deficits from upstream turbines are trans-ported in the streamwise direction, assuming Taylor’s hy-pothesis of frozen turbulence (Larsen et al., 2015). This setof assumptions allows for the decoupling of the wake deficitand wake-added turbulence formulations from the wake me-andering process (Larsen et al., 2007). Therefore, the threecomponents of the DWM model can be computed separatelyand successively superimposed on turbulence fields to gen-

erate wake time series, which can be used as inputs to aeroe-lastic simulations (Larsen et al., 2013; Keck et al., 2014a).

The wake deficit formulation of the DWM model is mainlybased on the work of Ainslie (1987) and solves the ax-isymmetric Navier–Stokes (N–S) equations with an eddyviscosity term and a set of calibration parameters. Initially,the DWM model was calibrated with CFD simulations per-formed by Madsen et al. (2010). Keck et al. (2012) deriveda two-dimensional model of the eddy viscosity term and up-dated the calibration parameters based on CFD simulations.Larsen et al. (2013) found that the calibration parameters ofthe former two studies were not suitable for predicting powerand loads at the Egmond aan Zee offshore wind farm. Tomatch the measured power, they introduced an artificial fil-tering function in the eddy viscosity term and re-calibratedthe deficit model; however, this calibration was not based onthe spatial description of the wake flow field but on powerproduction data. The eddy viscosity model to predict veloc-ity deficits in the current IEC standard (IEC, 2019) is inspiredby the work of Larsen et al. (2013).

Keck et al. (2014a, 2015) proposed a correction factor tothe eddy viscosity term, which includes the effects of atmo-spheric stability and shear on the turbulence mixing occur-ring in the wake, and re-calibrated the model parameters. Al-though these improvements were verified against large-eddysimulations (LESs), the influence of atmospheric stability onthe wake deficit evolution was hardly observed during a li-dar campaign (Machefaux et al., 2016; Larsen et al., 2015),in which it was argued that atmospheric stability affects to alarge extent the meandering process. A load validation studyusing the DWM model with calibrated parameters from bothMadsen et al. (2010) and Keck et al. (2012) as well as theIEC standard (IEC, 2019) was conducted by Reinwardt et al.(2018), who collected load measurements at the ECN windturbine test site in Germany and at the Technical Universityof Denmark (DTU) test site in Høvsøre in Denmark. Theyfound fatigue load biases within the range of 11 %–15 % forthe tower bottom and 8 %–21 % for the blade-root flapwisebending moments. Reinwardt et al. (2020) derived a new setof calibration parameters based on full-field lidar observa-tions of the wake field from a wind farm in the south-east ofHamburg, Germany. They demonstrated that improved wakedeficit predictions can be obtained by calibrating the DWMmodel with nacelle-mounted lidars.

The fidelity of the simulated wake meandering dynam-ics also affects the accuracy of load predictions (Larsenet al., 2013; Conti et al., 2021). Modelling of the meander-ing process relies on a suitable stochastic turbulence fieldand definition of the large-scale turbulence structures. Larsenet al. (2008) and Trujillo et al. (2011) demonstrated that thelarge-scale eddies can be extracted from the incoming atmo-spheric turbulence field from local mast measurements. Al-ternatively, the wake meandering process can be simulatedthrough synthetic wind fields generated using stochastic tur-bulence models (i.e. the turbulence model by Mann, 1994)

Wind Energ. Sci., 6, 1117–1142, 2021 https://doi.org/10.5194/wes-6-1117-2021

D. Conti et al.: Probabilistic estimation of the Dynamic Wake Meandering model parameters 1119

and a definition of the large-scale eddies or by means ofLESs. Machefaux et al. (2015) showed that inconsistenciesbetween a Mann-based and LES-based meandering processcan arise due to differences in the input turbulence fields.Larsen et al. (2008) and Trujillo et al. (2011) defined thelarge-scale eddies on the order of two rotor diameters (D)or larger as responsible for wake meandering, whereas otherstudies defined scales larger than 3–4D as dominant (Es-pana et al., 2011; Muller et al., 2015; Yang and Sotiropoulos,2019). Despite the severe impact of the wake meandering dy-namics on load predictions, its uncertainty has not been as-sessed in load validation studies due to lack of data (Larsenet al., 2013; Churchfield et al., 2015; Reinwardt et al., 2018).However, aeroelastic simulations with constrained wake me-andering dynamics can potentially decrease the uncertaintyin load predictions under wake conditions (Conti et al.,2021).

Further, the added turbulence formulation in the DWMmodel accounts for additional mechanically generated turbu-lence caused by the wake shear and the breakdown of tip androot vortices. These contributions are modelled by a semi-empirical formulation that uses parameters, which were cal-ibrated against CFD simulations (Madsen et al., 2010). Toour knowledge, no further development has been made onthis subject.

1.2 Problem statement

As described above, there is no consensus for the values ofthe DWM model parameters when studying load predictionsat any given site. Also, and perhaps most importantly, wedo not know the sources of uncertainty observed in previ-ous studies that used the model (Larsen et al., 2013; Church-field et al., 2015; Reinwardt et al., 2018), which need to beaddressed to provide reliable load predictions. The commonpractice has been to derive optimized sets of model param-eters based on limited synthetic or experimental data. Thishas led to an unknown confidence in the overall model pre-diction ability; incorrect calibration of the model parametersmay impact significantly the model performance and lead tosuboptimal wind turbine designs.

To address this issue, we estimate uncertainties in the cali-bration parameters of the DWM model by applying Bayesianinference (Box and Tiao, 1973), which consists of updatingany related prior information on model parameters by in-corporating new knowledge obtained from wake flow char-acteristics derived through lidar measurements. Further, theBayesian calibration provides a systematic approach to in-clude various types of uncertainty such as physical variabil-ity as well as measurement and modelling errors. This paperfocuses on improving and validating the calibration of DWMmodel parameters using lidar-derived wake features and hasa fourfold primary purpose:

1. Derive wake flow features such as the two-dimensionalvelocity deficit and wake-added turbulence profiles aswell as time series of the wake meandering in bothlateral and vertical directions from the SpinnerLidarmeasurements under different inflow wind speeds andatmospheric-stability conditions.

2. Calibrate the DWM-model-based wake deficitand wake-added turbulence predictions using theSpinnerLidar-derived wake flow features and theBayesian inference framework.

3. Propagate modelling uncertainties in fully resolvedwake flow fields for robust predictions that take into ac-count the calibrated uncertainties.

4. Conduct a sensitivity analysis to determine the most sig-nificant sources of uncertainty in simulated wake fieldsthat are typically inputs to aeroelastic simulations.

This study contributes to the ongoing discussion regardingthe accuracy of power and load predictions of wind turbinesoperating under wake situations (Conti et al., 2020b, 2021)by quantifying uncertainties in wake simulations performedwith the DWM model under a variety of inflow wind condi-tions. The outcomes of this study are useful for improvingcurrently adopted wake simulation procedures for load anal-ysis in the IEC standards as well as to provide practical rec-ommendations for wake model calibration studies based onmeasurements from nacelle-mounted lidars.

The work is organized as follows. Section 2 describes theDWM model. The SWiFT layout and relative wind site con-ditions are described in Sect. 3. In Sect. 4, we present thewind field retrieval assumptions used to derive wake featuresfrom SpinnerLidar measurements. The Bayesian calibrationof the DWM model is performed in Sect. 5. We carry outthe validation of the wind turbine wake simulations and con-duct a sensitivity analysis to investigate the most influentialparameters in Sect. 6. Finally, the last two sections are dedi-cated to the discussions and conclusions.

2 Dynamic Wake Meandering model

The DWM model resolves three main wake features: thequasi-steady velocity deficit, the wake-added turbulence, andthe wake meandering. Each model component is describedseparately in the following subsections.

2.1 Quasi-steady velocity deficit

The quasi-steady velocity deficit component describes thewake expansion and recovery caused partly by the recoveryof the rotor pressure field and partly by turbulence diffusionmoving farther downstream of the rotor (Larsen et al., 2013).The wake deficit is formulated in the meandering frame of

https://doi.org/10.5194/wes-6-1117-2021 Wind Energ. Sci., 6, 1117–1142, 2021


reference (MFoR), which is a coordinate system with originin the centre of symmetry of the deficit.

In the far-wake region, i.e. distances larger than two rotordiameters (Sanderse, 2015), the deficit evolution is assumedto be governed by turbulent mixing and is described by thethin shear layer approximation of the rotational symmetricN–S equations with the pressure term disregarded (Madsenet al., 2010). To account for the neglected pressure gradi-ent effects, an initial wake deficit is analytically formulatedbased on the turbine’s axial induction derived from blade el-ement momentum (BEM) theory (Madsen et al., 2010). Theturbulence closure of the N–S equations is obtained by meansof an eddy viscosity term, and the momentum equation issolved numerically using a finite difference scheme with theartificial initial deficit as a boundary condition (Madsen et al.,2010). Here, we use the numerical scheme of the stand-aloneDWM model (Liew et al., 2020; Larsen et al., 2020). We re-fer to the generalized definition of the non-dimensional eddyviscosity term by Keck et al. (2012), who considered two ma-jor drivers to the turbulence mixing: the ambient turbulence(TIamb) and turbulence induced by the wake shear layer:

νT

UambR(r, x)= F1(x)k1TIamb+F2(x)k2max(

Rw(x)2

UambR

∣∣∣∣∂U (x, r)∂r

∣∣∣∣ ; Rw(x)R

(1−

Umin

Uamb

)), (1)

where νT is the eddy viscosity, Uamb is the ambient windspeed at hub height, and R is the rotor radius. The first termon the right-hand side of Eq. (1) describes the contributionof the ambient turbulence and the second the self-generatedturbulence by the wake shear layer. Madsen et al. (2010) pro-posed the wake radiusRw(x), where x is the downstream dis-tance normalized byR, and the maximum velocity difference(Uamb−Umin), where Umin is the minimum wind speed inthe wake, as the turbulent length and velocity scales, respec-tively, that govern turbulent mixing due to the wake shearlayer.

Based on classical mixing length theory, Keck et al. (2012)defined the turbulence stresses to be proportional to the lo-cal velocity gradient ∂U (x, r)/∂r , which provides a two-dimensional eddy viscosity formulation that is a function ofthe axial and radial coordinates, x and r , respectively. Themax operator is included to avoid underestimating the turbu-lent stresses at locations where the velocity gradient of thedeficit approaches zero. Both terms in Eq. (1) include a fil-ter function (F1(x) and F2(x)) and a model constant (k1 andk2). The filter functions are required to model the turbulencedevelopment behind the rotor and have values in the range of0–1, depending on the downstream distance only (Keck et al.,2012). F1 accounts for the delay of the ambient turbulenceentrainment into the wake and is assumed to “activate” am-bient turbulence effects at downstream distances where thepressure has recovered (i.e. 2D downstream, where the farwake begins; Sanderse, 2015).

F2 compensates for the initial non-equilibrium betweenthe mean velocity field and the turbulent energy content cre-ated due to the rapid change in mean flow gradients close tothe rotor. We refer to Eqs. (17) and (18) in Keck et al. (2015)for the mathematical formulation of F1 and F2; k1 and k2are calibration parameters that govern the turbulence mixingand presumably do not change with wind turbine design andambient conditions (Keck et al., 2012).

2.2 Wake turbulence

The wake turbulence is composed of three turbulence sourcesand can be defined as follows (Vermeer et al., 2003):

TIwake =

√TI2

amb+TI2m+TI2

add, (2)

where TIm denotes the turbulence induced by the meander-ing of the wake deficit, and TIadd is the wake-added tur-bulence. TIm is commonly denoted as the apparent turbu-lence (Madsen et al., 2005) as the stochastic meandering ofthe wake deficit induces additional velocity fluctuations intotime series taken at fixed locations in the wake. This termis considered the main source of added turbulence in the farwake (Madsen et al., 2010), while its spatial distribution canbe computed by the convolution of the wake deficit in theMFoR and the probability distribution function (PDF) of thewake meandering in the lateral and vertical directions (Kecket al., 2014a). TIadd accounts for the shear- and mechanicallygenerated turbulence due to blade tip and root trailing vor-tices. The inhomogeneity of the wake-added turbulence ismodelled by scaling the local turbulence using the factor kmt(Madsen et al., 2010) as

kmt(r)=| 1−Udef,MFoR(r) | km1+

∣∣∣∣∂Udef,MFoR(r)∂r

∣∣∣∣km2, (3)

where Udef,MFoR is the velocity deficit in the MFoR, andkm1 and km2 are constants calibrated based on CFD results(Madsen et al., 2010). The wake-added turbulence derivedfrom Eq. (3) is presumed to meander together with the wakedeficit, thus being displaced by the large-scale eddies in theatmosphere.

2.3 Meandering model

Here, the meandering model is confined to a single wakescenario, whereas multiple wake dynamics are described inMachefaux (2015). The wake field is modelled by consid-ering a cascade of consecutive wake deficits that are dis-placed by the large-scale lateral- and vertical-velocity fluc-tuations, i.e. the wake transport velocities (vc and wc), cor-responding to the lateral (y) and the vertical axis (z), respec-tively. Adopting Taylor’s hypothesis, the downstream advec-tion of these deficits is assumed to be controlled by the meanwind speed of the ambient wind field. Larsen et al. (2008)estimated vc and wc by low-pass filtering of atmospheric



turbulence fluctuations. They defined a filtering cut-off fre-quency fcut,off = Uamb/(2D), thus excluding contributionsfrom smaller eddies to the meandering dynamics. This as-sumption was verified using full-scale lidar-based measure-ments collected behind an operating turbine (Bingöl et al.,2010). The wake displacements are computed as

y(x, t)= vc(t)t(x)+hyaw(x, t)

z(x, t)= wc(t)t(x)+htilt(x, t), (4)

where t = x/Uamb defines the time for an air particle to movefrom the rotor to the downstream distance in the wake region.An appropriate choice of the transport velocity of the wakeadvection lies between the ambient wind speed and the centrevelocity of the wake deficit (Keck et al., 2014b; Machefauxet al., 2015). The contribution from the yaw misalignment,which can redirect wakes in the lateral direction, is accountedfor by hyaw(x, t)= x tan(θ (t)), where θ (t) is the yaw offset atthe specific time (Machefaux, 2015; Vollmer et al., 2016).The contribution of the rotor tilt is considered by htilt(x, t)(Machefaux, 2015).

3 The SWiFT facility

The SWiFT facility is a research site located in Lubbock,Texas, operated by Sandia National Laboratories (Hergeset al., 2017). The site includes three Vestas V27 wind tur-bines, two meteorological towers, and a SpinnerLidar (Peñaet al., 2018) mounted on the nacelle of one of the turbines andlooking backwards. The entire site is on a fibre optic data ac-quisition and control network that synchronizes recordingsfrom masts, turbines, and the SpinnerLidar (Herges et al.,2017, 2018). The measurement campaign took place between2016 and 2017 with the main objective of characterizingwake fields and investigating wake steering control strategies(Herges et al., 2017).

Figure 1 provides an overview of the test layout togetherwith the notation used throughout the paper. In this study, weanalyse data collected at the meteorological mast (METa1),the turbine (WTGa1), and the SpinnerLidar mounted on thenacelle of the WTGa1. The METa1 (hereafter referred to asthe mast) is 60 m tall and instrumented with sonic anemome-ters at 10, 18, 32, 45, and 58 m, sampling at 100 Hz. Other in-struments installed on the mast are reported in Herges et al.(2017). The mast is placed 2.5D south of WTGa1 in com-pliance with the IEC standard guidelines (IEC, 2015, 2017).As southerly winds are prevalent at the site (see Fig. 1,right), this layout allows the retrieval of concurrent incom-ing wind conditions from METa1, wake measurements be-hind the WTGa1 performed by the SpinnerLidar, and powerand load measurements on the waked WTGa2 installed 5Ddownstream. The WTGa1 and WTGa2 are variable-speedand pitch-regulated turbines with a hub height of 32.1 m,D = 27 m, a cut-in wind speed of 3 m/s, and a maximumpower output of 192 kW reached at the rated wind speed

of 12 m/s (Berg et al., 2014). The supervisory control anddata acquisition (SCADA) is available for both turbines, pro-viding records of the rotor speed, pitch and yaw angles, andpower production, among other information, at 50 Hz.

3.1 SpinnerLidar

The SpinnerLidar is a research Doppler wind lidar devel-oped at DTU based on a continuous-wave (CW) laser system(Peña et al., 2018). Hereafter, SpinnerLidar and lidar denotethe same system. The SpinnerLidar has been mounted eitherin the spinner or on top of the nacelle of a wind turbine (An-gelou and Sjöholm, 2015; Peña et al., 2018). In this study,the SpinnerLidar was installed on the nacelle of the WTGa1and scanned the rotor wake at a high temporal and spatialresolution so that wake features could be derived. For theSWiFT campaign, the SpinnerLidar scanned continuously ina rose-pattern every 2 s (see Fig. 2), and the system inter-nally subdivided the rose into 984 sections. The accumu-lated Doppler-shifted spectra at each of the sections was alsorecorded (Herges et al., 2017).

Once a scan was completed, the SpinnerLidar refocusedat a different range, and this process took about 2 s (Hergeset al., 2018). For the SWiFT campaign, several scanningstrategies were adopted as described below.

– Strategy I. The SpinnerLidar scanned seven downstreamdistances: 1, 1.5, 2, 2.5, 3, 4, and 5D. A full cycle (i.e.from 1 to 5D) took 30–42 s. This dataset is suitablefor investigating the wake deficit evolution and recov-ery behind the rotor; however, the frequency is too lowto properly derive turbulence estimates or meanderingdynamics.

– Strategy II. The SpinnerLidar scanned at the fixed dis-tance of 2.5D, ensuring both high spatial and temporalresolution; ≈ 298 rosette scans were generated within a10 min period. This dataset is suited for turbulence andmeandering investigations.

– Strategy III. The SpinnerLidar scanned at the fixed dis-tance of 5D behind the rotor, generating about 298scans each 10 min. During this period, power and loadmeasurements were recorded on WTGa2. This datasetis suitable for load validation analysis. Since it providesa description of the wake flow field, including velocitydeficits, turbulence, and meandering at a distance thatcorresponds to typical spacings in wind farms, it is avaluable dataset for validating fully resolved wake flowpredictions as long as induction effects are accountedfor.

3.2 Site conditions

For extended periods of the campaign, WTGa1 operated un-der large yaw misalignment as wake steering strategies were



Figure 1. (a) A sketch of the SWiFT layout that includes locations of the main devices (i.e. wind turbines, masts, and the SpinnerLidar). Theshaded red area indicates that the SpinnerLidar scans in the wake of WTGa1 assuming winds from the south. The distances are normalizedwith the rotor diameter D. (b) The wind rose at the site derived from the 32 m sonic observations collected on METa1 during the campaign.

Figure 2. A schematic view of the SpinnerLidar’s scanning pattern:(a) a front view at 2.5D in the wake; (b) a top view including allscanned distances behind the WTGa1, which is depicted by solidblue lines. The WTGa2 is also shown.

being investigated (Herges et al., 2017). To consider periodswhere WTGa1 is nearly aligned with the mean inflow, wefiltered out 10 min periods characterized by an average yawoffset larger than±10◦ compared to the free-stream wind di-rection (Conti et al., 2020a). The yaw offset is here defined asthe difference between the nacelle orientation and the winddirection measured at the mast. Further, we focus the anal-ysis on periods for which the free-stream wind direction iswithin 90–270◦ (thus southern winds; see Fig. 1, right). Thisleads to about 850 available 10 min periods.

Figure 3 shows 10 min statistics of the hub-height turbu-lence intensity (TIamb), the power-law shear exponent (α),and the power production of WTGa1 as a function of thehub-height mean wind speed (Uamb) based on the mast in-flow measurements; α is computed from the sonic measure-ments at 18 and 45 m. As shown, the site is characterized bya wide range of turbulence and shear conditions, which are aconsequence of the varying atmospheric stability (Doubrawa

et al., 2019; Conti et al., 2020a). Further, relatively low windspeeds are recorded (3–10 m/s); thus WTGa1 operates belowrated power, as seen in Fig. 3c. Because of this range of op-erating conditions, high rotor thrust coefficients that inducestrong wake deficits characterize this dataset.

3.2.1 Atmospheric stability

Here, we investigate the variability in the wake flow charac-teristics under varying stability and inflow wind speed condi-tions. We classify each 10 min sonically derived statistic intoatmospheric-stability classes defined by ranges of the dimen-sionless stability parameter (z/L), where L is the Obukhovlength (Monin and Obukhov, 1954) computed from the sonicmeasurements as

L=−u3∗T

kgw′2v′, (5)

where u∗ =√−u′w′ is the friction velocity, u′w′ is the lo-

cal kinematic momentum flux, k = 0.4 is the von Kármánconstant, g is the acceleration due to gravity, T is the meansurface-layer temperature, the vertical velocity componentis denoted by w, and 2v is the virtual potential tempera-ture (which we approximate by the sonic temperature). Theprime denotes fluctuations around the mean value, and theoverbar is a time average. We define three main atmospheric-stability classes based on z/L ranges by Peña (2019): unsta-ble (−2< z/L <−0.2), near-neutral (−0.2< z/L < 0.2),and stable (0.2< z/L < 2) atmospheric conditions. We usethe measurements at the 18 m sonic anemometer to derivethe stability within each 10 min period. As shown in Contiet al. (2020a), the sonic measurements at 18 m provide thebest fit to the polynomial form of Högström (1988), which



Figure 3. Inflow wind and operational conditions at the SWiFT site. (a) Hub-height turbulence intensity as a function of the hub-heightmean wind speed based on the mast inflow measurements, (b) power-law shear exponent derived using observations from the 18 and 45 msonic measurements, and (c) power productions of WTGa1 recorded from SCADA. Each marker represents a 10 min period.

describes the relation between the dimensionless wind shearφm and the dimensionless stability parameter z/L in the sur-face layer (see middle panel of Fig. 3 in Conti et al., 2020a).

3.3 Data statistics

The statistics of the inflow wind parameters are presentedseparately in Tables 1, 2, and 3 according to the relative Spin-nerLidar scanning strategy. Table 1 presents data collectedduring Strategy I. There is a fair number of 10 min periodsto characterize the variability in the wake deficit with respectto atmospheric stability, inflow wind speeds, and downstreamdistances. The table shows increasing turbulence levels underunstable compared to stable cases, whereas relatively highvertical wind shears are found under stable conditions, as ex-pected. The dataset is thus suitable for analysing the effectsof atmospheric stability on the wake recovery. The datasetcollected during Strategy II is reported in Table 2 and is usedto characterize wake turbulence and meandering under dif-ferent stability conditions. For Strategy III, represented in Ta-ble 3, the dataset is characterized by stable conditions mainlyas the records correspond to night hours within three consec-utive nights in July 2017.

4 Lidar measurement processing

As lidars only measure the line-of-sight (LOS) velocity(vlos), assumptions are needed to reconstruct the three-dimensional wind field u= (u,v,w), where u is the lon-gitudinal, v the lateral, and w the vertical velocity com-ponent. If we neglect any probe volume averaging alongthe beam, vlos depends on the unit directional vector n=

(cosφ cosθ,cosφ sinθ,sinφ), which describes the scanninggeometry through the elevation (φ) and azimuth (θ ) anglesand the wind field u,

vlos(φ,θ )= ucos(φ)cos(θ )+ v cos(φ) sin(θ )+w sin(φ). (6)

Considering the small elevation angles and the typical lowvalues ofw, we assumew = 0 (Doubrawa et al., 2019, 2020).

Table 1. Dataset from Strategy I. The data are classified accordingto wind speed bins of 1 m/s and three atmospheric-stability classes:stable (s), near-neutral (nn), and unstable (u). The number of 10 minsamples is also indicated; α is the power-law shear exponent; TIambis the turbulence intensity defined as the standard deviation of hori-zontal wind speed divided by the mean wind speed. The wind speedand turbulence parameters are obtained from sonic observations at32 m height.

U Samples α TIamb[m/s] [–] [–] [%]

s nn u s nn u s nn u

3± 0.5 5 3 6 0.39 0.36 0.08 7 10 184± 0.5 19 4 11 0.30 0.10 0.01 8 19 225± 0.5 25 5 13 0.27 0.13 0.01 7 11 226± 0.5 30 8 23 0.28 0.15 0.04 7 11 167± 0.5 13 12 16 0.23 0.12 0.02 7 12 138± 0.5 6 9 4 0.27 0.10 0.04 7 12 109± 0.5 5 12 3 0.30 0.17 0.02 7 11 9

Table 2. Similar to Table 1 but for Strategy II.



5± 0.5 2 4 12 0.16 0.07 0.01 7 14 126± 0.5 – 1 8 – 0.04 0.01 – 13 127± 0.5 9 – 8 0.22 – 0.10 10 – 148± 0.5 3 5 1 0.18 0.12 0.05 10 12 14

This assumption may introduce an error of up to 3 % inthe reconstructed horizontal wind speed at short distances(1–2D) (Debnath et al., 2019). Following the approach ofDoubrawa et al. (2020), we can combine the u and v veloc-ity components into a total horizontal wind vector, U , andEq. (6) becomes

vlos(φ,θ,θ0)= U cos(φ)cos(θ − θ0), (7)



Table 3. Similar to Table 1 but for Strategy III.



4± 0.5 1 – – 0.38 – – 7 – –5± 0.5 2 – – 0.32 – – 7 – –6± 0.5 18 – – 0.30 – – 6 – –7± 0.5 50 – – 0.25 – – 8 – –8± 0.5 24 2 – 0.21 0.04 – 8 14 –9± 0.5 2 2 – 0.18 0.02 – 10 12 –

where θ0 is the yaw offset, and the overbar indicates asmoothed signal as we apply a moving average operatorwith a 15 s window to the yaw misalignment to accountfor any temporal delay from the spatial distances amongthe mast, turbine’s nacelle, and SpinnerLidar measurements(Conti et al., 2020a). With Eq. (7), we can reconstruct hor-izontal wind velocity measures at each individual scannedpoint within the rosette pattern. Further, we linearly interpo-late the reconstructed wind speeds across the rosette patterninto a two-dimensional regular grid with a 2 m resolution,which is sufficient to characterize the spatial characteristicsof the wind field in wakes (Fuertes et al., 2018; Conti et al.,2020a).

4.1 Lidar-estimated wake deficit

To perform comparisons with predicted velocity deficits fromthe DWM model, we aim at isolating the contribution of thewake deficit from that of the vertical wind shear in lidar mea-surements. As defined in Trujillo et al. (2011), the quasi-instantaneous wake deficit profile can be obtained by sub-tracting the mean vertical shear profile (Uamb(z)) from thequasi-instantaneous wake recording as

Udef(x,y,z)=Uamb(z)−U (x,y,z)

Uamb(z), (8)

where U (x,y,z) is estimated from lidar measurements us-ing Eq. (7), and Uamb(z) is the relative 10 min average in-flow vertical wind speed profile measured at the mast. Thedeficit is then normalized with respect to the ambient windspeed profile. The vlos measurements and also the recon-structedU wind velocities are defined on a coordinate systemthat is attached either to the nacelle (nacelle frame of refer-ence, NFoR), which rotates with the yawing of the turbine,or to the ground (fixed frame of reference, FFoR). To per-form direct comparisons with the DWM model predictions,the lidar-estimated deficits obtained from Eq. (8) need to becomputed in the MFoR. Here, this is performed by trackingthe wake centre position through the method of Trujillo et al.(2011), where a bivariate Gaussian shape is fitted to the ve-locity deficit flow field, and the wake centre is the geometric

centroid of the Gaussian function:

fdef =A

2πσwyσwzexp

[−

12

((yi −µy)2

σ 2wy

+(zi −µz)2

σ 2wz

)], (9)

where µy and µz define the wake centre location; σwy andσwz are width parameters of the wake profile in the y andz directions, respectively; yi and zi denote the spatial loca-tions of the lidar measurements; andA is a scaling parameter.Each scanned point of the quasi-instantaneous wake record-ing can be translated into the MFoR using the estimated µyand µz from Eq. (9) (Reinwardt et al., 2020). Therefore, wecan compute the multiple wake recordings within a 10 minperiod in the MFoR and subsequently compute flow statis-tics such as the ensemble-average deficit profile as well as thespatial distribution of the wake turbulence in the MFoR. Toensure a high-quality fit, we reject scans where the estimatedwake centre location is within ≈ 10% of the lateral boundsof the scanning area and at more than 0.75D from the hubheight in the vertical direction (Doubrawa et al., 2020; Contiet al., 2020a).

Figure 4 illustrates ensemble-average measured deficitprofiles in the MFoR at 2, 3, 4, and 5D behind the rotorobtained from all 10 min periods characterized by an incom-ing wind speed of 7 m/s and under varying stability regimesduring Strategy I (see Table 1 for reference). We can clearlyobserve the impact of the atmospheric stability and in partic-ular of the associated turbulence levels on the wake recov-ery behind the rotor. A strong and well-defined symmetricwake deficit shape is seen under stable conditions (top row),whereas the deficits recover faster moving downstream as theatmosphere becomes more unstable (bottom row).

4.2 Lidar-estimated wake turbulence

Turbulence measures derived from lidar radial velocity mea-surements are “filtered” because of their relatively largeprobe volume (Peña et al., 2017), and so they are generallylower than those obtained from sonic observations. Never-theless, if the Doppler spectrum of the vlos is available, wecan potentially circumvent the averaging effects and estimatethe unfiltered variance of vlos (Peña et al., 2017; Mann et al.,2010). Mann et al. (2010) assume that the ensemble-averagedDoppler spectrum over a time period 〈S(vlos)〉 is related to theprobability distribution of the vlos at the focus distance andcan be computed as

〈S(vlos)〉 =

∞∫−∞

ϕ(s)p(vlos|s)ds, (10)

where ϕ(s) is the spatial averaging function of the lidar thatdepends on the position along the beam s, and p(vlos|s) de-notes the PDF of vlos at the location s. If we assume thatthe PDF of vlos is independent of s, (i.e. there is no ve-locity gradient along the beam), then Eq. (10) reduces to



Figure 4. Ensemble-average velocity deficit profiles in the MFoR measured at 2, 3, 4, and 5D behind the rotor for an inflow wind speedof 7 m/s under stable (upper row), near-neutral (middle-row), and unstable (lower row) conditions. The number of scans used to derive theensemble statistics ranges between 312 and 636, depending on data availability. The SpinnerLidar scanning pattern is shown by red dots,whereas the turbine rotor area is illustrated by solid blue lines. The vertical and lateral coordinates are normalized by the rotor radius andcentred at hub height.

〈S(vlos)〉 = p(vlos). As a result, the vlos statistics (i.e. meanand variance) can be computed from the first and second cen-tral moments of p(vlos) as

µvlos =

+∞∫−∞

vlosp(vlos)dvlos,

σ 2vlos=

+∞∫−∞

(vlos−µvlos )2p(vlos)dvlos, (11)

where µvlos and σ 2vlos

denote the mean and unfiltered vari-ance of vlos, respectively. Nevertheless, velocity gradientsalong the lidar beam may appear when measuring at the wakeedges, which can introduce errors in the estimated turbulence(Meyer Forsting et al., 2017). Following the procedure ofPeña et al. (2019), we compute the ensemble-averaged nor-malized Doppler spectrum within 10 min periods by thresh-olding the noise-flattened spectra with a value of 1.2 and cor-recting them by subtracting the background spectrum. Weaccumulate the LOS Doppler spectra onto the regular grid ofthe scanned area and estimate µvlos and σ 2

vlosfor each grid

cell using Eq. (11). As discussed in Herges and Keyantuo(2019), invalid measurements occur due to the boresight andground return as well as the return from the rotating rotor ofWTGa2, if in operation. These invalid observations appear as

a very high return signal in the Doppler spectrum in proxim-ity to low wind speeds (i.e. at approximately 1 m/s) and areremoved. The filtering effects due to the probe volume can bequantified by computing the ratio between filtered and unfil-tered LOS variances across the rosette pattern; we find ratiosin the range 0.8–0.9 at 2.5D, which vary according to stabil-ity conditions (not shown).

Examples of 10 min ensemble-averaged Doppler spectraobtained at three fixed locations across the scanned area –a wake centre, a wake edge, and a wake-free position – areshown in Fig. 5 for an incoming wind speed of 7 m/s andambient turbulence of 6 %. A narrow spectrum with a single-peak distribution centred at about 7 m/s for the wake-freelocation (green) is seen, whereas spectrum-broadening ef-fects induced by small-scale generated turbulence are notice-able for the positions within the wake. The wake centre (red)shows a wider spectrum with a peak at a significantly lowerwind speed than the incoming flow, whereas the wake edge(cyan) shows a double-peak distribution that may be partiallydue to the inhomogeneity of the wind field along the beam(Herges and Keyantuo, 2019) and also due to the meander-ing occurring within the analysed 10 min period.

To characterize the spatial distribution of the wake turbu-lence within the scanned area, we derive σ 2

U estimates di-



Figure 5. Examples of normalized Doppler LOS velocity spectra measured over a 10 min period at 2.5D in the wake at three differentlocations – wake centre (red), wake edge (cyan), and wake-free (green) – for an incoming wind speed of 7 m/s and ambient turbulence of6 %.

rectly by applying the variance operator to Eq. (7):

σ 2vlos= σ 2

U cos(φ)2 cos(θ − θ0)2, (12)

where σ 2U is the variance of the horizontal wind speed, and

as shown, covariance terms are neglected. As the LOS isalmost never aligned with the u velocity component acrossthe rosette, except at the centre of the pattern, σ 2

vloscan be

“contaminated” by the variances and covariances of the othervelocity components (Peña et al., 2017). Therefore, the re-lation in Eq. (12) can lead to inaccurate estimations of thelongitudinal-velocity variances. Peña et al. (2019) estimatedthe contamination of different components on the LOS vari-ances for the SpinnerLidar and showed that the ratio of theunfiltered LOS velocity variance to the variance of the longi-tudinal velocity component is generally lower than 1 acrossthe scanned area, except at the centre, where the ratio is1, and within an area above the centre, where it can behigher than unity. Although the adopted retrieval assumptionin Eq. (12) introduces uncertainties in the turbulence mea-sures, we can account for the expected errors in the Bayesianinference framework.

Figure 6 illustrates the spatial distribution of the unfilteredσ 2

U computed in the MFoR, normalized with the u-velocityvariance of the ambient wind field measured at the 32 msonic anemometer (σ 2

u, amb). Under stable conditions and fora downstream distance of 2.5D, we can observe an enhance-ment in turbulence levels in proximity to the rotor tips, es-pecially in the upper part of the rotor (see Fig. 6a). The ob-served added turbulence is caused by the breakdown of therotor tip vortices. These features are no longer noticed as theatmosphere becomes more unstable where a more uniformand less prominent distribution of the turbulence is found(see Fig. 6b and c).

5 Calibration of the DWM model in the MFoR

The calibration of the wake deficit and wake-added tur-bulence components are conducted in the MFoR using a

Bayesian inference framework. We describe the Bayesianmodel in Sect. 5.1 and provide calibration results for thewake deficit in Sect. 5.2 and for the wake-added turbulencein Sect. 5.3. We investigate wake meandering dynamics sep-arately in Sect. 5.4.

5.1 Bayesian inference formulation

The basis of the Bayesian inference is to estimate the prob-ability distribution of the model parameters based on avail-able observations. Let θm = {k1,k2, . . . ,km1,km2} be a set ofmodel parameters to be estimated using lidar-derived wakefeatures (i.e. wake deficit and wake-added turbulence profilesin the MFoR) denoted by yd = {yd1,yd2, . . . ,ydn}, where nis the number of available observations. We consider that theexperimental data and the model predictions satisfy the pre-diction error equation:

yd = g(θm,Xm)+ ε, (13)

where g(θm,Xm) denotes the DWM model predictions ob-tained from a particular set of model parameters (θm)and a set of observable variables (Xm). Here Xm =

{TIamb,Uamb,α,CT,x,y,z} includes the inflow wind condi-tions measured at the mast (TIamb,Uamb,α); the rotor thrustcoefficient of the turbine (CT), which is derived from theBEM model (Madsen et al., 2010); and the spatial locationsof the scanning pattern (x,y,z). ε = εy + εm denotes a ran-dom prediction error composed of two terms: the measure-ment error εy and the model prediction error εm. The formeris described by a zero mean normal distribution with standarddeviation σεy , which is determined from field observations.The latter is assumed to have zero mean, which implies un-biased model predictions, and a standard deviation σεm to bedetermined by the Bayesian estimation along with the modelparameters. To facilitate statistical inference, we assume Xmas deterministic inputs (i.e. free of uncertainty) and that themodel error εm is independent of the set of input variables



Figure 6. Two-dimensional spatial distribution of the horizontal wind velocity variance (σ 2U) derived in the MFoR at 2.5D in the wake,

normalized with the u-velocity variance of the ambient wind field (σ 2u, amb) for three 10 min periods characterized by (a) stable, (b) near-

neutral, and (c) unstable conditions. Approximately 298 scans of the wake are processed for each 10 min period. The relative ambient windspeed ranges between 6.5 and 8.5 m/s.

Xm and described by a normal distribution. This implies thatthe model predictions are normally distributed for a givenXm, which is a reasonable choice for wake deficit profiles.The Bayesian approach for model calibration deals with up-dating the combined parameter set (θm,σεm ), given a set ofobservations (yd,Xm) by applying the Bayes theorem:

f (θm,σεm |yd)=f (yd|θm,σεm )f (θm,σεm )

f (yd), (14)

where f (θm,σεm |yd) is the updated posterior distributionof the model parameters, f (θm,σεm ) is the prior dis-tribution that is typically assigned based on subjectiveor previous information, f (yd|θm,σεm ) denotes the likeli-hood of observing the data yd from a model with cor-responding θm parameters, and f (yd) is the prior pre-dictive distribution that is defined as the marginal distri-bution f (yd)=

∫f (yd|θm,εm)f (θm,σεm )dθmdεm. By using

the prediction error in Eq. (13) and assuming that the er-ror terms are jointly normal with a zero mean vector andcovariance matrix

∑εyd = diag(σ 2εyd

) and∑εm = diag(σ 2

εm),

the measured quantities follow the normal distribution yd ∼

N(g(θm,Xm|yd),

∑ε

), where the covariance matrix takes

the form∑ε =

∑εyd+∑εm . As a result, the likelihood func-

tion of observing the data follows the multi-variable normaldistribution defined as

f (yd|θm,εm)=

∣∣∑ε

∣∣−1/2

(2π )n/2exp

[−

12

[yd− g(θm,Xm|yd)

]T−1∑ε

[yd− g(θm,Xm|yd)

]], (15)

where the |.| denotes the determinant. The analytical and dif-ferentiable solution of the posterior distribution of the pa-rameters in the N–S equations with the eddy viscosity termof Eq. (1) is not readily available. Therefore, we employ anumerical sampling method to approximately evaluate theposterior distribution and its first and second moments. Here,the adaptive no-U-turn Markov chain Monte Carlo (MCMC)

sampler is employed to generate samples from the poste-rior distribution (Hoffman and Gelman, 2014; Salvatier et al.,2016).

The outcome of the calibration is a joint probability dis-tribution of the inferred model parameters. From this jointPDF, we can estimate the posterior PDF of any wake fea-ture simulated by the DWM model, i.e. the wake deficit andwake-added turbulence profiles in the MFoR or the fully re-solved wakes in the FFoR, among others, which we denoteby q:

f (q|yd)=∫2

f (q|θm)f (θm|yd)dθm. (16)

The posterior distribution of the wake feature q in Eq. (16)can be solved numerically using sampling methods (e.g.Monte Carlo simulations), so its first and second moment canbe estimated.

5.2 Wake deficit parameter estimation

We use lidar-derived wake deficit profiles in the MFoR col-lected during Strategy I and Strategy II and employ theBayesian model to infer uncertainty in the k1 and k2 pa-rameters of the eddy viscosity term in Eq. (1). These pa-rameters were found to be the most sensitive to the resultingwake deficit predictions (Keck et al., 2012). The predictionmodel of Eq. (13) is constructed as follows. The experimentaldata (yd) comprise two-dimensional ensemble-average lidar-estimated deficit profiles binned according to downstreamdistances (3, 4, and 5D), atmospheric stability (i.e. stable,near-neutral, and unstable), and wind speed bins of 1 m/s inthe range of 3–9 m/s. Note that we discard measurements inthe near wake (at 2D) for improving the quality of the fittingin the far-wake region (see Fig. 8).

The set of observable variables comprises Xm =

{TIamb,Uamb,α,CT,x,y,z}, where the inflow parameters(TIamb,Uamb,α) are provided in Tables 1 and 2; the rotorthrust coefficient CT is derived from the BEM model imple-



mented in the aeroelastic code HAWC2 (Larsen and Hansen,2007) and based on the aerodynamics and airfoil inputs ofthe SWiFT turbine (Doubrawa et al., 2020) (CT = 0.84, thatis nearly constant for wind speeds below 9 m/s); and x,y,and z refer to the spatial coordinates of the deficits resolvedin the MFoR. The uncertainties in measured deficit profilesare computed as εyd (r)= σ (r)/

√n, where σ (r) is the stan-

dard deviation of all 10 min deficits within the analysed caseat the radial position r , and n is the number of 10 min periods(also referred to as samples in Table 1).

We select uniform prior distributions on model parame-ters k1, prior ∼ U(0.001,0.2) and k2, prior ∼ U(0.001,0.2) onintervals that consider physical constraints, ensuring conver-gence of results and covering previous calibrations reportedin the literature. Thus, we employ an MCMC algorithm tosample from the posterior PDFs of the calibration parame-ters using the Bayesian framework. The inferred joint andmarginal posterior distributions of the model parameters (k1and k2) are shown in Fig. 7, together with point values fromearlier studies (Madsen et al., 2010; Keck et al., 2012; Larsenet al., 2013; IEC, 2019; Reinwardt et al., 2020). As shown,the lidar-based wake deficits are informative, and we obtainwell-defined posteriors that follow a normal distribution withk1 ∼N (0.081,0.017) and k2 ∼N (0.015,0.003). The nega-tive correlation between k1 and k2 seen in Fig. 7 indicatesthe interdependence of the physically induced effects as bothparameters contribute to turbulence diffusion. It is found thatthe posterior means of the informed parameters k1 and k2differ from those recommended in the IEC standard (IEC,2019). Generally, low k1 and k2 values attenuate the degreeof turbulence mixing in the wake, which lead to strong ve-locity deficits persisting at distances farther downstream. Weprovide the statistical properties (mean, variance, and coeffi-cient of variation) of the inferred parameters in Table 4 andcorrelation measures in Table 5.

5.2.1 Wake deficit predictions

We propagate the uncertainties in k1 and k2 to predict wakedeficits in the MFoR and compare them with the ensemble-average lidar-derived profiles in Fig. 8. First, we observe thatthe lidar-estimated deficits exhibit a faster wake recovery asthe atmosphere becomes more unstable compared to stableregimes. This effect is mainly caused by the enhanced tur-bulence mixing occurring under unstable conditions as theyare characterized by ambient turbulence levels 2–3 timeshigher than those of the stable cases (see Table 1). The lidar-observed maximum deficit varies between 30 % and 60 %within the first five rotor diameters, depending on the inflowturbulence conditions. Similar behaviours were reported inrecent lidar measurement campaigns (Iungo and Porté-Agel,2014; Machefaux et al., 2016; Fuertes et al., 2018; Zhanet al., 2020b).

The lidar-estimated deficits are approximately Gaussianunder stable to near-neutral conditions, whereas the Gaussian

shape is lost under more unstable conditions. This may resultfrom errors in the wake tracking procedure due to the largermeandering amplitudes and also due to the presence of large-scale turbulence structures in the inflow (Conti et al., 2020a).The rotor thrust is another factor governing the variability inthe wake recovery (Zhan et al., 2020b); however, its influ-ence is secondary for the dataset analysed here due to therelatively low incoming wind speeds and relatively constantthrust coefficients (Conti et al., 2020a).

The DWM-model-predicted deficit profiles with param-eters specified by their posterior distributions are in goodagreement with the lidar observations for distances beyond4D (see Fig. 8). For these distances, the turbulence mix-ing effects dictated by the ambient and self-generated waketurbulence on the deficit recovery are fairly well capturedby the inferred parameters. The nominal model predictionsgenerally fit the observations, whereas an overlap betweenthe measurements and the region of modelling uncertaintyis found. The largest deviations between predicted and mea-sured deficits are found at shorter distances (2–3D) and aremainly due to the model inadequacy to simultaneously fit allthe experimental measurements and experimental uncertain-ties. The assumptions adopted to describe the near-wake re-gion also introduce uncertainty in the deficit predictions atshort distances (Keck et al., 2015; Machefaux et al., 2016).

It can be observed that the uncertainties in k1 and k2 pa-rameters primarily influence the depth of the wake (i.e. themaximum deficit), while the sensitivity to these parametersdecreases significantly with the outer radial distance. It isalso noticed that the uncertainty in the deficit predictions in-creases for high ambient turbulence and far downstream dis-tances. This is because k1 is proportional to TIamb in Eq. (1),and the sensitivity of the model parameters increases as thewake recovers. To provide a measure of the uncertainty inwake deficit predictions, we compute the coefficient of vari-ation COV = σ/µ, where σ is the standard deviation, andµ is the mean value of the maximum deficit, obtained bypropagating the PDFs of k1 and k2, as in Eq. (16). We findCOV= 3 % under stable conditions (Tamb = 0.07), whichincreases to 6 % under unstable conditions (Tamb = 0.14),for an incoming wind speed of 7 m/s. This result confirmsthat the uncertainty in wake deficit predictions increases forhigher turbulence, but it also shows that uncertainties in k1and k2 parameters do not lead to uncertainty of the samemagnitude in deficit predictions resolved in the MFoR (forreference, COVk1 = 21 % and COVk2 = 19 %, as reported inTable 4).

We provide comparisons between measured and predictedwake deficit profiles using the calibration from this work aswell as those reported in early studies in Fig. 9. For this par-ticular analysis, we analyse predictions at 5D behind therotor for an inflow wind speed of 7 m/s under stable, near-neutral, and unstable regimes. The main discrepancy amongthe models is the relative sensitivity of the wake recoveryto the ambient turbulence. This is primarily governed by k1;



Figure 7. Joint and marginal posterior PDFs of k1 and k2 parameters. The uncertainty regions representing the 10 %, 30 %, 66 %, and95 % confidence intervals are shown. The histograms obtained from 40 000 MCMC samples and the corresponding empirical PDFs are alsoincluded. The calibration parameters from early studies are shown with red markers and discussed in the text.

however, in the eddy viscosity model of Larsen et al. (2013),IEC (2019), and Reinwardt et al. (2020), it also depends on anonlinear coupling function Famb(TIamb) that attenuates thewake recovery for turbulence above ≈ 12% (see Fig. 6 inLarsen et al., 2013). This function was introduced to fit thepower productions at the Egmond aan Zee offshore windfarm, and it is not based on observations of the wake field(Larsen et al., 2013). The wake recovery predicted with themodels of Larsen et al. (2013) and IEC (2019) is practicallyinsensitive to ambient turbulence rising from 7 % to 16 % atdownstream distances up to 5D. Similar outcomes are re-ported in Fig. 13 in Reinwardt et al. (2020), who showedthat the model of Larsen et al. (2013) provided conservativedeficits for ambient turbulence up to 16 %.

By considering k1 and k2 as universal constants (Kecket al., 2012), the quality of the dataset utilized for the modelcalibration is essential to ensure reliable parameter estima-tion. As previous calibrations were carried out on larger ro-tors than those of the SWiFT turbines and utilized eitherpower production data (Larsen et al., 2013; IEC, 2019) orlimited CFD simulations (Madsen et al., 2010; Keck et al.,2015) and one-dimensional scans of the wake by a nacellelidar (Reinwardt et al., 2020), these aspects may explain theobserved deviations in Fig. 9.

5.3 Improved wake-added turbulence formulation

The wake-added turbulence model (Eq. 3) assumes that tur-bulent structures (i.e. tip and root vortices) are unaffectedby atmospheric turbulence. The rotor-induced vortices are

rapidly disrupted under high-turbulence conditions, causingthe breakdown within the first 2D (Madsen et al., 2005).However, vortices can persist and extend at farther distancesunder low to moderate turbulence combined with stable strat-ification conditions (Ivanell et al., 2009; Subramanian et al.,2018; Conti et al., 2020a). Thus, the wake-added turbulenceprofiles can exhibit a more pronounced double-peak featurein the proximity of the rotor tips or a more uniform distri-bution depending on the atmospheric turbulence conditions(this effect is also seen in Fig. 6). Equation (3) also assumesradially symmetric wake-added turbulence profiles. How-ever, the inflow vertical wind shear re-distributes the waketurbulence. Enhanced turbulence levels are actually observedin the proximity of the upper tip of the rotor blade (Vermeeret al., 2003; Chamorro and Porte-Agel, 2009; Conti et al.,2020a). Figure 6a shows this effect.

Due to these two assumptions, we propose an improvedsemi-empirical formulation of the wake-added turbulencescaling factor k∗mt, which produces wake profiles in betteragreement with the lidar observations. This is achieved byrelating both the depth and the velocity gradient terms to theambient turbulence and by including the effect of the inflowvertical wind shear on the vertical-velocity deficit gradient as

k∗mt(y,z)=| 1−Udef(y,z) | (k∗m1TIamb+ k∗

q1)

+

∣∣∣∣∂U∗def(y,z)∂y∂z

∣∣∣∣ (k∗m2TIamb+ k∗

q2), (17)

where U∗def(y,z)= (U (z)Udef(y,z))/max (U (z)Udef(y,z)),and k∗m1,k

∗

q1,k∗

m2, and k∗q2 are parameters to be determined



Figure 8. Comparison between measured and predicted ensemble-average spanwise velocity deficit profiles resolved in the MFoR at hubheight and obtained at 2, 3, 4, and 5D behind the rotor (from left to right) and for inflow wind speeds ranging from 3 to 8 m/s with a 1 m/sbin (from top to bottom panel). The SpinnerLidar-measured (markers) and DWM-model-predicted (solid lines) deficits are shown for eachstability class (stable in blue, near-neutral in green, and unstable in red). The error bars represent the measurement uncertainty, while theshaded areas represent the uncertainty in the model predictions; both sources of uncertainty refer to the 95 % confidence interval (see Table1 for details on the inflow wind conditions).

using Bayesian inference. Figure 10 illustrates the two-dimensional profiles of the depth and velocity deficit gradi-ent terms of Eqs. (3) and (17). As illustrated, by includingthe term U∗def(y,z), we obtain a turbulence field that mimicsqualitatively well the observed enhanced turbulence withinthe upper wake region.

5.3.1 Estimation of wake-added turbulence parameters

The calibration parameters in Eq. (17) are inferred basedon the lidar-estimated wake-added turbulence profiles inthe MFoR collected during Strategy II. For this particulardataset, the SpinnerLidar scans at a fixed distance of 2.5D,ensuring about 298 scans for every 10 min period. The in-flow characteristics are reported in Table 2. As the Doppler



Figure 9. Ensemble-average spanwise velocity deficit profiles computed in the MFoR at hub height obtained at 5D behind the rotor for anincoming wind speed of 7 m/s under stable (blue), near-neutral (green), and unstable (red) atmospheric conditions. The measured turbulenceintensities are 7 %, 12 %, and 16 %, respectively. The SpinnerLidar-measured profiles are shown by markers and their relative 95 % confi-dence interval by the error bars. The DWM-model-predicted deficits are shown by solid lines; each panel refers to model predictions usingcalibration parameters from a number of studies (see text for more details). The “SpinnerLidar” panel refers to the model proposed in thecurrent study, while the shaded areas indicate its 95 % confidence interval.

Figure 10. DWM-model-predicted flow characteristics. (a) Velocity deficit, (b) velocity deficit term with combined vertical shear profileUdef(y,z)∗, (c) gradient of the velocity deficit, (d) gradient of the profile resulting from the combined velocity deficit and the vertical shear.The flow characteristics are computed for Uamb = 6 m/s, TIamb = 7 %, and α = 0.25 at a downstream distance of 2.5D.

LOS velocity spectrum is available, we derive unfiltered LOSvariances as described in Sect. 4.2 and subsequently the tur-bulence intensity as the ratio of the standard deviation to themean of the horizontal wind speed. From Eq. (2), we canisolate TIadd (wake-added turbulence term) by firstly resolv-ing the wake recordings in the MFoR, which eliminates thecontribution of TIm, and then by subtracting TIamb, which isderived from the 32 m sonic observations in front of the rotor.

As a first step, we fit the “original” analytical formula-tion of kmt from Eq. (3) to each individual lidar-estimatedwake-added turbulence profile to estimate the values of km1and km2 using a simple least-squares optimization algorithm.Figure 11a and d show the relation between the estimatedoptimal parameters (in markers) and the ambient turbulence.It is shown that km1 increases, and km2 decreases almost lin-early for increasing turbulence intensity. This indicates thestrong effect of the deficit gradient term (proportional to km2in Eq. 3) under low-turbulence conditions, which both am-plifies the double-peak feature of the wake turbulence profileat the rotor tips and mimics the wake vortices. As the ambi-ent turbulence increases, km1 and lower km2 become larger,which indicates a more uniform distribution of the wake tur-bulence. These effects are observed in Fig. 11b and c (see

markers), which show the spanwise distribution of the lidar-derived wake-added turbulence for two 10 min periods withrelatively low and high values of ambient turbulence inten-sity, 7 % and 12 %, respectively. We also compare the mea-sured and predicted vertical distribution of TIadd in Fig. 11eand f, which shows that slightly improved predictions can beobtained using Eq. (17), i.e. considering the effects of the at-mospheric shear on wake turbulence.

To infer the posterior PDFs of k∗m1,k∗

q1,k∗

m2, and k∗q2,we assign prior distributions based on the linear depen-dencies observed in Fig.11a and d, namely km1, prior ∼

U(0,6), kq1, prior ∼ U(0,0.1), km2, prior ∼ U(−50,10), andkq2, prior ∼ U(0,5) as well as k1 ∼N (0.081,0.017) and k2 ∼

N (0.015,0.003) previously estimated in Sect. 5.2. The un-certainties in lidar-derived wake-added turbulence profilesaccount for errors introduced by the flow modelling assump-tions of Eq. (12), which neglects cross-contamination ef-fects on the LOS variance. We define these errors as zero-mean normally distributed with standard deviation σεyd

=

0.1 · Tadd(y,z), which leads to a coefficient of variation of10 % (Peña et al., 2019).

The resulting posterior PDFs of the parameters follow anormal distribution (not shown) with statistical properties



tabulated in Table 4. We provide the mean values of theinferred k∗m1,k

∗

q1,k∗

m2, and k∗q2 values in the form of a lin-ear regression model that relates the depth and deficit gradi-ent terms of Eq. (17) to the ambient turbulence in Fig. 11aand d. The shaded area represents the 95 % confidence in-terval, which is obtained by propagating the PDFs of theturbulence-related and wake deficit parameters. As shown,the predictions are characterized by a relatively high degreeof uncertainty. This is because, for example, there are fewavailable observations to characterize turbulence; the mea-surement uncertainties are relatively high; and the modellingsimplifications, such as the semi-empirical k∗mt, introduce fur-ther uncertainties. We provide the correlations among all theinferred parameters obtained from the joint PDF in Table 5.As expected, the wake deficit parameters k1 and k2 are neg-atively correlated to k∗m1 and k∗q1 as k∗m1 and k∗q1 are propor-tional to the wake deficit term in Eq. (17), which is the maindriver to the intensity of the wake-added turbulence.

5.4 Wake meandering

Here, we investigate the relationship between the inflow tur-bulence fluctuations and the lidar-tracked wake positions tocharacterize the large-scale eddies responsible for the mean-dering. The analysis is carried out by comparing the spectraof the lidar-tracked meandering time series, which are de-rived by means of the tracking algorithm in Eq. (9), with thatsimulated from the meandering model in Eq. (4), where vcandwc are obtained from the sonic observations at 32 m. Yawmisalignment from SCADA is also included. We conduct theanalysis using the dataset collected at 2.5D behind the rotorduring Strategy II and classify all the available 10 min peri-ods according to stability to derive ensemble-average spectrafrom multiple observations with similar inflow conditions.Note that the measured inflow wind speeds are below rated,and turbulence levels from all 10 min periods within each sta-bility class are similar to those reported in Table 2.

Results of the spectral analysis for both lateral and ver-tical meandering are shown in Fig. 12. Here, the ensemble-average spectra from the SpinnerLidar observations are com-pared to those from the meandering model without low-pass filtering the incoming turbulence fluctuations (denotedas DWMwf). The spectra are normalized with their relativevariances and plotted as a function of the commonly usedStrouhal number St = fD/U∞, where f denotes frequency,and U∞ is the aggregated wind speed from the ensemble-average statistics. As shown, the slope of the lidar-basedspectra (red lines) matches that of DWMwf (blue lines) upto St = 0.3–0.5, which corresponds to three and two rotordiameters, respectively. For St > 0.5, the energy content ofthe lidar-estimated spectra remarkably decreases comparedto that of DWMwf. These observations indicate that large-scale turbulent structures (>2D) are dominant in the wakemeandering (Trujillo et al., 2011; Heisel et al., 2018). Whencompared to the stable case, the spectra under unstable condi-

tions show a higher energy content at large turbulence scalesor equivalently low Strouhal number.

Figure 12 shows that a stochastic description of the large-scale eddy size might be appropriate. Thus, we describe thelarge-scale eddies responsible for the wake meandering byintroducing the stochastic variable Dm, which is normallydistributed with a mean equal to 2.5D (it corresponds to thewake diameter at 5D behind the rotor) and a standard devi-ation of 0.3D based on the observations in Fig. 12. The re-sulting 95 % confidence intervals are shown as shaded areasin Fig. 12. The uncertainty in Dm is found negligible whencomputing wake meandering time series; this is shown in Ap-pendix A.

6 Validation of the DWM model in the FFoR

The validation of the DWM model is performed by resolv-ing wake fields in the FFoR; thus the simulated wakes in-clude the combined effects of the velocity deficit, added tur-bulence, and wake meandering dynamics in both lateral andvertical directions. This analysis is carried out using datafrom Strategy III, i.e. at a fixed distance of 5D behind therotor, ensuring a sufficient number of scans to derive unfil-tered turbulence estimates as well as wake meandering timeseries within a 10 min period. The analysed dataset is pri-marily characterized by stable conditions, as seen in Table 3,i.e. low turbulence intensities (6 %–8 %) and strong verticalshears (α = 0.18–0.38).

6.1 Correction for rotor induction effects

The SpinnerLidar measurements collected during Strat-egy III were taken in the induction zone of the WTGa2(Herges et al., 2018). For induction correction, we em-ploy the two-dimensional induction model of Troldborg andMeyer Forsting (2017), which accounts for both longitudinaland radial variation in the induced wind velocity:

Cind =[1− a0

(1−

ξx√1+ ξ2

x

)·

(2

exp(+βaεa)+ exp(−βaεa)

)2], (18)

where a0 is the induction factor at the rotor centre area,a0 = 0.5(1−

√1− γaCT); γa = 1.1, ξx = x/R is the distance

in front of the rotor normalized by the rotor radius; ρa =√y2+ z2/R denotes the radial distance from the rotor cen-

tre axis; and εa = ρa/√λa(ηa + ξ2

x ), βa =√

2, λa = 0.587,and ηa = 1.32 (Troldborg and Meyer Forsting, 2017; Dim-itrov, 2019). Note that only lidar measurements taken acrossthe rotor area of WTGa2 are corrected by the induction fac-tor in Eq. (18). The estimated induction factors indicate thatthe wind speed can be reduced at hub height by up to 12 %upstream of the WTGa2 and below rated power (see alsoFig. 14).



Figure 11. Wake-added turbulence predictions. Panels (a) and (d) show the relation of both km1 and km2 in Eq. (3) to the ambient turbulence.The linear regression model that is determined using Bayesian inference is shown by solid lines, whereas the shaded areas indicate the95 % confidence interval obtained by propagating the posterior PDFs of the parameters. The comparison between measured (markers) andpredicted (lines) lateral wake-added turbulence profiles resolved in the MFoR at hub height and obtained at 2.5D behind the rotor is shownin (b) and (c) for TIamb = 7 % and 12 %, respectively, whereas the relative vertical profiles are shown in (e) and (f). The error bars indicatethe measurement uncertainty, whereas the shaded areas indicate that of the model predictions relative to the 95 % confidence interval. Thepredictions from Eq. (3) are computed using the mean values of km1 = 0.11 and km2 = 0.54, which are obtained from all the observations in(a) and (d).

6.2 Uncertainty propagation of simulated wake fields

We derive the two-dimensional spatial distribution of themean wind speed in the wake region (UFFoR) by the convolu-tion between the wake deficit in the MFoR (Udef,MFoR) andthe PDF of the meandering path (fm) (Keck et al., 2015):

UFFoR(y,z,k1,k2,Dm,ym, ε,zm, ε)=

Uamb

(z

zhub

)α ∫ ∫Udef,MFoR(y− ym+ ym, ε,z− zm

+ zm, ε,k1,k2) · fm(ym,zm,Dm,ym, ε,zm, ε)dymdzm, (19)

where ym and zm denote the spatial coordinates of the wakemeandering time series, and ym, ε and zm, ε are measures oftheir relative uncertainties. We introduce these errors to ac-count for incorrect wake tracking positions that can arise dueto the adopted wake tracking algorithm; ym, ε and zm, ε are as-sumed to be uncorrelated and to follow a normal distributionwith zero mean and standard deviation such that the 95 %percentile corresponds to approximately 4 m, which is twicethe resolution adopted to interpolate SpinnerLidar measure-ments onto the regular grid (see Sect. 4). Note that the at-mospheric shear profile Uamb(z) is superposed after the wakedeficit calculation (Madsen et al., 2010). Similarly, the two-dimensional spatial distribution of the u-velocity variance

(σ 2uFFoR

) can be computed as

σ 2uFFoR

(y,z,k1,k2,k∗

m1,k∗

q1,k∗

m2,k∗

q2,Dm,ym, ε,zm, ε)=

σ 2uamb+

∫ ∫ ((UMFoR(y− ym+ ym, ε,z− zm+ zm, ε,k1,k2)

−UFFoR(y,z,k1,k2,Dm,ym, ε,zm, ε))2+(k∗mt,MFoR (y

−ym+ ym, ε,z− zm+ zm, ε,k1,k2,k∗

m1,k∗

q1,k∗

m2,k∗

q2)

·UMFoR(y− ym+ ym, ε,z− zm+ zm, ε,k1,k2))2)

· fm(ym,zm,Dm,ym, ε,zm, ε)dymdzm, (20)

where UMFoR = Uamb(z)Udef,MFoR, UFFoR is derived fromEq. (19), and σ 2

uambis the variance of the ambient u velocity

component. Alternatively, UFFoR and σ 2uFFoR

can be equiva-lently derived by superposing the wake deficit and the wake-added turbulence factor k∗mt on stochastic turbulence fieldswith constrained meandering path and subsequently by com-puting the first- and second-order statistics of the syntheticwind fields. However, the analytical forms in Eqs. (19) and(20) can be easily used to propagate the posterior PDFs of thecalibration parameters to predict profiles of UFFoR and σuFFoR

with relative uncertainties. The inflow parameters (Uamb, α,and σ 2

uamb) in Eqs. (19) and (20) are required inputs for the

DWM model and are estimated from mast measurements.



Figure 12. Normalized ensemble-average power spectral density (PSD) of the lateral and vertical wake meandering tracked by the Spinner-Lidar (red) and that derived using the meandering model (DWMwf; shown in blue) under stable (column a), near-neutral (column b), andunstable conditions (column c). The ensemble-average PSDs are computed for data collected at 2.5D behind the rotor and are normalizedwith their relative variances. The 95 % confidence interval in the large-scale-eddy definition is shown by the grey area (see text for moredetails).

Table 4. Mean (µ), standard deviation (σ ), and coefficient of variation (COV= σ/µ) estimated from the posterior PDFs of model parameters.The values of k1, k2, σεdef , km1, kq1, km2, kq2, and σεadd are determined using Bayesian inference. Dm denotes the spatial size of the large-scale eddies governing wake meandering dynamics, and ym, ε and zm, ε denote the wake tracking position errors expressed in metres.

Modules Wake deficit Wake-added turbulence Wake meandering

k1 k2 σεdef k∗m1 k∗q1 k∗m2 k∗

q2 σεadd Dm ym, ε [m] zm, ε [m]

µ 0.081 0.015 0.02 1.33 −0.02 −5.61 1.09 0.03 2.5 0 0σ 0.017 0.003 0.001 0.14 0.014 0.97 0.09 0.001 0.3 1.5 1.5COV 21 % 19 % 5 % 10 % 59 % 28 % 10 % 3 % 12 % – –

We compute UFFoR and σuFFoR from Eqs. (19) and (20)by constraining the meandering path (fm) using either thelidar-tracked meandering time series, which we denote asDWM∗, or by using the meandering model in Eq. (4) withlow-pass-filtered vc- and wc-velocity fluctuations obtainedfrom the 32 m sonic anemometer and the yaw misalign-ment of WTGa1 obtained from SCADA, which we denote asDWM∗∗. In the latter model, the low-pass-filtered frequencyis defined as a function of the stochastic variable associatedwith the size of the large-scale eddies (Dm), as discussed inSect. 5.4. Figure 13 shows the comparison between observedand predicted (with both DWM∗ and DWM∗∗ models) pro-files of the mean wind speed and its standard deviation ob-tained from two different 10 min periods.

A good agreement between measurements and predictionsis found. The vertical profile of the mean wind speed exhibitsa single-peak shape resulting from the combined effects ofthe inflow vertical shear (modelled by a power law) and thewake-induced Gaussian-like deficit shape. The wake turbu-lence in the lateral direction exhibits a double-peak shapewith larger values near the locations associated with strongvelocity gradients that are further enhanced by the wake me-andering. Enhanced turbulence levels in proximity of the up-per wake region are observed from lidar measurements. Thedeviations between DWM∗ and DWM∗∗ model predictionsare more pronounced for the simulated turbulence than forthe wind speed fields and are exclusively due to differencesin the meandering representations.



Table 5. Correlation coefficients between model parameters estimated using Bayesian inference.

k1 k2 σεdef k∗m1 k∗q1 k∗m2 k∗

q2 σεadd

k1 1 −0.73 <0.01 −0.15 −0.02 0.02 −0.06 −0.06k2 −0.73 1 <0.01 −0.12 <0.01 <0.01 −0.01 <−0.01σεdef <0.01 <0.01 1 <0.01 <0.01 <0.01 <0.01 <0.01k∗m1 −0.15 −0.12 <0.01 1 −0.61 −0.21 −0.57 −0.01k∗q1 −0.02 <0.01 <0.01 −0.61 1 0.19 −0.26 0.06k∗m2 0.02 <0.01 <0.01 −0.21 0.19 1 −0.47 −0.03k∗q2 −0.06 −0.01 <0.01 −0.57 −0.26 −0.47 1 −0.03σεadd −0.06 <0.01 <0.01 −0.01 0.06 −0.03 −0.03 1

The wake simulation uncertainties shown in Fig. 13 aredetermined by propagating uncertainties in model parame-ters (Table 4) and by accounting for relative correlations (Ta-ble 5) using Monte Carlo simulations. The uncertainties inlidar-measured wind speeds account for volume averagingeffects and for errors introduced by the retrieval assumptions(e.g. w sin(φ)= 0 m/s) (Debnath et al., 2019). The uncertain-ties in lidar-derived turbulence (here defined as the standarddeviation of the wind speed) account for errors introducedby neglecting cross-contamination effects in Eq. (12) (Peñaet al., 2019).

To evaluate the performance of the DWM model, we cal-culate two flow metrics that are relevant in aeroelastic sim-ulations and compare them to relative measured quantities:the rotor-effective wind speed (Ueff), defined as the weightedsum of wind speeds across the rotor area, and the effectivewake turbulence (σu, eff), which is derived as the weightedsum of turbulence estimates across the rotor. Figure 14 showsthe one-to-one comparison between measured and DWM∗-and DWM∗∗-model-predicted flow metrics for all 10 min pe-riods. We find a slope that deviates < 1 % from unity, withR2= 0.95 for Ueff and a bias of ≈ 4 % and R2

≈ 0.93 forσu, eff. The observed scatter is explained by the large mea-surement uncertainties, by the uncertainties in inflow windparameters, and by those of the model predictions. The latteris estimated by propagating uncertainties in model parame-ters provided in Table 4, which result in a COV of 1 % forUeff, and of 3 % for σu, eff. These findings indicate that theinferred uncertainties in model parameters do not propagateinto an error of the same magnitude on the fully resolvedwakes that are eventually input to aeroelastic simulations.

Further, a sensitivity analysis indicates that the variationsin Ueff and σu, eff are mostly explained by the uncertainties ink1 and k2 and the wake tracking position ym, ε,zm, ε ; this isshown in Appendix B. The contribution of the wake-addedturbulence from Eq. (17) to the total wake turbulence σu, effin the FFoR is marginal and accounts for approximately 2 %–7 %. The turbulence induced by the meandering of the wakedeficit is thus the major source of added turbulence (Madsenet al., 2010).

7 Discussion

The SpinnerLidar measurements show that the wake recov-ers faster under unstable compared to stable conditions pri-marily due to the high turbulence levels of the former (Iungoet al., 2013; Zhan et al., 2020b). The wake recovery can bepredicted accurately by the DWM model when using appro-priate calibration parameters. Further, accurate reproductionof the wake meandering dynamics in both lateral and verticaldirections is key for accurate wake simulations.

Although the SWiFT campaign provides a comprehensivedataset, including a wide range of inflow wind conditions anddetailed two-dimensional lidar measurements of the wakefield, the calibration parameters obtained from the 192 kWturbine with 32 m diameter should be further evaluated formulti-megawatt turbines with larger rotors. While this studycannot explicitly demonstrate the transferability of the ob-tained results for modern-size turbines, it highlights the needfor datasets that include observations of the wake deficit pro-files under varying stability conditions, inflow wind speeds,and downstream distances for the reliable and robust calibra-tion of engineering wake models. Here, we find that the ve-locity deficit’s recovery rate for increasing turbulence is themain difference among calibrations reported in the literature(see Fig. 9), which were based on either power productiondata (Larsen et al., 2013; IEC, 2019) or limited CFD sim-ulations (Madsen et al., 2010; Keck et al., 2015) and one-dimensional observations of the wake field from a nacellelidar (Reinwardt et al., 2020).

We also demonstrate that uncertainties in calibration pa-rameters (e.g. COVk1 ≈ 21 % and COVk2 ≈ 19 %) do notlead to uncertainty of the same magnitude in the wind speedand turbulence fields (e.g. COVUeff ≈ 1 % and COVσu, eff ≈

3 %) that are eventually used as input for the aeroelasticsimulations. Note that the uncertainty in the simulated wakefields increases for increasing ambient turbulence due to theproportionality of k1 to TIamb in Eq. (1). Although a differentdataset might produce other calibration parameters, it doesnot necessarily imply an improved accuracy of the simulatedwake fields in the FFoR. Particularly, we expect our calibra-tion and that of Madsen et al. (2010), Larsen et al. (2013),



Figure 13. Comparisons between SpinnerLidar (SL)-measured and DWM-model-predicted spatial distribution of the mean and standarddeviation velocity computed in the FFoR and obtained at 5D in the wake region for two 10 min periods. The predictive DWM model thatincorporates the SpinnerLidar-tracked meandering time series is denoted by DWM∗ (red line), and that based on the meandering modelcomplemented by measured inflow turbulence fluctuations from the mast and yaw offsets is denoted by DWM∗∗ (blue line). The solid linesrepresent the mean predictions, whereas the shaded areas indicate the 95 % confidence intervals. The uncertainties in measurements areshown with error bars. The top view profiles are centred at hub height, whereas the side view profiles are centred along the vertical symmetryplane of the wake.

Figure 14. Comparison of the SpinnerLidar-measured (SL) and DWM-model-predicted rotor-effective wind speeds Ueff (a) and rotor-effective turbulence σu, eff (b). The DWM∗ predictions (red markers) are obtained by constraining the meandering on the basis ofSpinnerLidar-tracked wake displacements, while DWM∗∗ uses mast-based meandering time series (blue markers). The SpinnerLidar Ueffstatistics derived by neglecting induction effects are shown by grey markers (see Sect. 6.1 for more details).

and IEC (2019) to provide similar wake fields in the FFoRfor low ambient turbulence (i.e. ≤ 7 %), as illustrated by theblue lines in Fig. 9.

We recommend conducting DWM model calibrationsusing two-dimensional high-temporal- and high-spatial-resolution measurements of the wake field. Such resolu-tion can be achieved by research-based nacelle lidars today,

which allow the resolution of wake flow features includingthe wake deficit and wake-added turbulence profiles in theMFoR for model calibration. When using power productiondata for such calibrations (the IEC standard values are basedon this approach), we are unable to distinguish between un-certainties from inaccurate wake deficit predictions and thosefrom erroneous wake meandering representations. The latter



play an important role in the accuracy of the fully resolvedwake fields, which are inputs to aeroelastic simulations.

As wind turbines are typically spaced 5D and beyond, itis recommended to focus future measurement campaigns andmodel validation studies on including those regions. Addi-tional high-resolution datasets will allow a more comprehen-sive validation analysis to verify that the model proposed inthis work is applicable for a wide range of conditions (e.g.turbine types and site conditions). To this extent, Bayesianinference is a valuable approach for updating the PDFs ofmodel parameters estimated within this work by directly in-cluding data from future observations while retaining infor-mation from the earlier observations. Further, power and loadvalidation analyses using the proposed calibration parame-ters at multiple sites can increase the confidence in our cali-bration methodology.

Characterizing wake turbulence using lidars is challeng-ing due to the limited sampling frequency and probe volumeeffects (Peña et al., 2017). We demonstrate the usefulnessof Doppler radial velocity spectra to compute unfiltered LOSvariances in wake conditions. In addition to the enhanced tur-bulence intensity, the wake turbulence is characterized bothby being highly isotropic and reduced turbulence length scalecompared to the ambient turbulence (Madsen et al., 2005).These turbulence characteristics were not investigated in thiswork.

8 Conclusions

We analysed high-spatial- and high-temporal-resolutionSpinnerLidar measurements of the wake field collected atthe SWiFT facility and derived wake features such as thewake deficit, wake-added turbulence, and wake meander-ing under varying atmospheric-stability conditions, inflowwind speeds, and downstream distances. The SpinnerLidar-estimated wake characteristics computed in the MFoR wereused to determine uncertainties in the DWM model param-eters using Bayesian inference. The uncertainties in modelparameters were propagated to predict fully resolved wakeflow fields in the FFoR. This approach allowed us to quantifyuncertainties in the DWM-model-simulated wake fields andto investigate the sensitivity of the model parameters to flowfeatures that primarily affect power and load predictions.

The SpinnerLidar-derived wake deficit profiles revealedthe strong impact of atmospheric stability on wake evolu-tion. In particular, we observed the faster recovery of thedeficit under unstable compared to stable regimes as higherturbulence intensities characterized the former. These effectswere accurately reproduced by the eddy viscosity term of theDWM model with the inferred parameters for distances be-yond 4D. Our results indicate that the currently adopted pa-rameters in the IEC standard lead to conservative velocitydeficit predictions (up to 18 % for moderate to high ambientturbulence TIamb ≥ 12 %) at distances up to 5D behind therotor.

We proposed and verified an improved semi-empirical for-mulation of the wake-added turbulence model that capturedthe effects of the atmospheric shear and the ambient turbu-lence on the spatial re-distribution of the wake turbulenceobserved at 2.5D. We also demonstrated that the wake me-andering is the major source of added turbulence in the wakeregion.

The underlying hypothesis of the DWM model (i.e. wakesare advected passively by the large eddies in the incom-ing wind field) was verified by means of the SpinnerLidar-tracked meandering time series. The spectral analysis indi-cated that large eddies associated with sizes larger than 2Dare responsible for the wake meandering; however, the large-eddy “definition” had only marginal effects on the predictedwake fields. Accurate tracking of the wake centre positionwas the most influential factor in simulating wake flow fieldsaccurately. We expect it to also play a central role in the ac-curacy of power and load predictions.

In a future study, we will quantify uncertainties in powerand load predictions based on the proposed calibration at twodifferent sites, the SWiFT facility and the Nørrekær Engewind farm in Denmark (Conti et al., 2020b).



Appendix A: Comparisons of wake meandering timeseries

Figure A1 compares measured and predicted time series ofthe wake meandering in the lateral direction under varyingstability conditions. The DWM model predictions are com-puted by applying the filtering cut-off frequency with thestochastic definition of the large-scale eddies (Dm). A rea-sonable agreement between the two signals is found, wherethe major wake displacements are captured by the meander-ing model. Note that improved correlation can be achievedby utilizing a reduced advection wind speed in the time-lagparameter in Eq. (4) compared to the ambient wind velocity.Nevertheless, the largest observed movements are induced bythe yawing of the WTGa1, which is fairly frequent within theanalysed dataset. Further, the 95 % confidence interval of themeandering predictions obtained by propagating uncertain-ties in the Dm definition is nearly negligible. This indicatesthat uncertainties in the filtering cut-off frequency used inthe DWM model formulation have a marginal effect on theaccuracy of wake simulations.

Figure A1. Time series of the wake meandering in the lateral direction observed by the SpinnerLidar (red markers) and those derivedfrom the meandering model of Eq. (4) denoted as DWMfilt. (blue lines) under stable (a), near-neutral (b), and unstable conditions (c). Thepredictions are obtained at 2.5D behind the rotor. The shaded blue areas indicate the 95 % confidence intervals obtained by propagating theuncertainty in the large-scale-eddy definition



Appendix B: Sobol sensitivity indices

Sensitivity analysis is conducted to identify the most im-portant parameters affecting the accuracy of wake simula-tions. Here, we only investigate the uncertainties in the modelparameters, which are listed in Table 4, and assume thatthe inflow conditions are perfectly prescribed. We employa variance-based sensitivity method and compute total Sobolindices (Sobol, 2001; Saltelli et al., 2010). The Sobol sensi-tivity decomposes the variance of the response (e.g. Ueff andσU, eff) into contributions from input parameters and associ-ated interactions.

The Sobol indices computed from wake simulations at fiverotor diameters behind the rotor are illustrated in Fig. B1.Note that uncertainties in wake centre locations (ym, ε andzm, ε) have a similar influence as for the calibration parame-ters (i.e. k1 and k2) on the predicted flow features such asUeffand σu, eff. Overall, tracking the wake meandering in both lat-eral and vertical directions is of primary importance in wakefield representations and therefore in power and load vali-dations. The sensitivity of the wake-added turbulence modelparameters as well as the large-scale-eddy definition (Dm) ismarginal. It is inferred that these parameters can be consid-ered to be deterministic without compromising the accuracyof wake simulations.

Figure B1. Variance decomposition (Sobol indices) for the rotor-effective wind speed (a) and a measure of the rotor-effective turbulence (b).



Data availability. The SpinnerLidar and meteorological mast datacan be requested from the authors.

Author contributions. DC, ND, AP, and TH participated in theconception and design of the work. DC conducted the data analysisand wrote the draft manuscript. ND and AP supported the overallanalysis and critically revised the manuscript. TH contributed withthe acquisition of the dataset and critically revised the manuscript.

Competing interests. The authors declare that they have no con-flict of interest.

Disclaimer. Publisher’s note: Copernicus Publications remainsneutral with regard to jurisdictional claims in published maps andinstitutional affiliations.

Acknowledgements. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technologyand Engineering Solutions of Sandia, LLC., a wholly owned sub-sidiary of Honeywell International, Inc., for the US Department ofEnergy’s National Nuclear Security Administration under contractDE-NA-0003525.

Review statement. This paper was edited by Sandrine Aubrunand reviewed by two anonymous referees.

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Probabilistic estimation of the Dynamic Wake Meandering